The present invention relates to a piezoelectric resonator, vibrating in a contour mode, having a characteristic curve of variation of frequency as a function of temperature of the cubic type and capable of being produced by chemical etching of a quartz substrate.
A contour mode resonator has the form of a thin plate with displacement in its plane. Its thickness must be sufficiently small for the forces of inertia caused by movement outside the plane to have a negligible effect on the deformation energy. The most frequently adopted form is a rectangle which entails the existence of four geometric parameters, namely:
two angles of cut defining the direction of the normal to the plate;
one angle of cut defining the orientation of the sides of the rectangle in the plane of the plate, and
a dimensional ratio between the sides of the rectangle.
For a resonator to be of practical interest, it is desireable for its first order thermal coefficient to be close to 0 at the average utilization temperature.
Differently cut quartz crystals are on the market, the GT cut being the one which has the most favorable thermal properties. It is a rectangular plate the dimensional ratio of which is equal to 0.86, and which is obtained by rotation about the electric axis X of the crystal followed by rotation +45.degree. about the normal. The GT-cut resonator oscillates in a contour mode and more particularly in the elongation mode in the small dimension of the plate. The first and second order thermal coefficients are nil, and the third order coefficient is very small. The disadvantage of this cut arises from the fact that the thermal properties of the resonator depend critically on the dimensional ratio of the plate.
By way of example, for a GT-cut quartz crystal the first order thermal coefficient .alpha. is equal to .+-.0.1.times.10.sup.-6 /.degree.C., the second order coefficient .beta. is equal to .+-.1.times.10.sup.-6 /.degree.C..sup.2 and the third order coefficient .gamma. is lower than 30.times.10.sup.-12 /.degree.C..sup.3.
However, it should be noted that for a relative variation in the dimensional ratio w/l, equal to 1%, the variation of the first order thermal coefficient is equal to 2.5.times.10.sup.-6 /.degree.C. This means that the thermal coefficient of a GT-cut quartz crystal inevitably has to be adjusted after assembly of the resonator.
Another known cut is the DT cut, which consists of a generally square plate obtained by rotation about the electric axis X and vibrating in a surface shear mode. DT-cut resonators have the advantage over GT-cut resonators of being insensitive to variations in the dimensional ratio. However, their thermal qualities are less advantageous. By way of example, the first order thermal coefficient .alpha. is nil, the second order thermal coefficient .beta. is equal to -(15 to 20).times.10.sup.-9 /.degree.C..sup.2 and the third order thermal coefficient .gamma. is approximately equal to 45.times.10.sup.-12 /.degree.C..sup.3.
AT-cut quartz crystals are also known, which are in the form of a plate obtained by rotation about the electric axis X of the crystal. The data for these quartz crystals are given in particular in the publication "Quartz Vibrators and Their Applications" by Pierre Vigoureux, published by His Majesty's Stationary Office, London 1950. Two types of AT-cut quartz crystals oscillating at a frequency of 4 MHz are currently on the market. These are the AT quartz by Nihon Dempa, of which the thermal properties at 25.degree. C. are as follows:
first order thermal coefficient: PA1 second order thermal coefficient: PA1 third order thermal coefficient: PA1 first order thermal coefficient: PA1 second order thermal coefficient: PA1 third order thermal coefficient:
.alpha.=.+-.0.1.times.10.sup.-6 /.degree.C.; PA2 .beta.=4.+-.1.times.10.sup.-9 /.degree.C..sup.2 ; PA2 .gamma.=95.times.10.sup.-12 /.degree.C..sup.3 ; PA2 .alpha.=.+-.0.1.times.10.sup.-6 /.degree.C.; PA2 .beta.=(11.+-.1).times.10.sup.-9 /.degree.C..sup.2 ; PA2 .gamma.=90.times.10.sup.-12 /.degree.C..sup.3 ;
and the AT quartz by the Societe Suisse pour L'Industrie Horlogere (SSIH), of which the thermal properties are 25.degree. C. are as follows:
In addition to the thermal properties inferior to those of GT-cut quartz crystals, AT-quartz crystals have a frequency four times higher for comparable bulk. On the other hand, the first order thermal coefficient is more sensitive to differences in the value of the angle of cut. For example, for a variation .phi. in the angle .phi. equal to 1.degree., the corresponding variation .alpha. in the first order thermal coefficient is equal to 4.7.times.10.sup.-6 /.degree.C. Moreover, the Nihon Dempa AT quartz has a complicated form, being bevelled at each end of the bar with the lateral faces being inclined. This results in the need for individual metallization after completion of machining. The SSIH AT quartz is very long, i.e., in the region of 11 mm.
Another equally well known resonator, with good thermal properties and quasi-independence of the first order thermal coefficient relative to the dimensional ratio w/l, is the ZT-cut quartz described in French Pat. No. 2,435,855. Such a resonator, the basic form of which is a rectangle, may also have a more complex structure by combining several basic rectangles which, from the point of view of the propagation of elastic waves, may be considered pseudo-free rectangles. Examples of such combinations may be found in the afore-mentioned French Patent and in French Pat. No. 2,521,782, which describes ZT-cut embeddable resonator structures.
ZT-cut resonators are obtained from a Z-cut substrate, that is a plate with the optical axis Z of the quartz crystal as its normal by a first rotation about the mechanical axis Y of the crystal, followed by a second rotation about the normal to the plane of the resonator. The need for two rotations may cause certain disadvantages. In particular, the second rotation in the plane of the substrate involves oblique orientation of the resonators with respect to the edges of the substrate and, consequently, non-optimum utilization thereof. Furthermore, when resonators are cut by chemical etching of the substrate, the lateral faces perpendicular to the direction of the vibration wave are oblique, rather than perpendicular, to the plane of the substrate. This obliqueness of the lateral faces may cause troublesome effects for which it is difficult to compensate. It has, in particular, the effect of favoring coupling between the desired contour mode and parasitic vibration modes outside the plane.